Problem 20: This coming December, “Tron: Legacy” will be released to theatres in

Question

Problem 20: This coming December, “Tron: Legacy” will be released to theatres in both 2D and 3D. I do not like watching movies in 3D (both because the 3D glasses don’t fit over my glasses very well, and because I’m a grumpy old man who hates change). Suppose that in the Seattle area, if a movie theatre is showing “Tron: Legacy”, there is a 25% chance that it will be showing it in 2D. Also assume that each theatre’s decision of whether or not to show it in 2D is independent of every other theatre’s decision.

a) If I know that eight theatres in the area will be showing the movie, what is the mean and standard deviation for the number of theatres that will be showing it in 2D?

b) What is the probability that at least one theatre out of the 8 will show it in 2D?

c) Suppose I want to call the theatres to find out whether they will be showing it in 2D. I make a list of the theatres, in random order, and start calling them. I go through the list until I find a theatre that is showing it in 2D, at which point I stop calling. What is the probability I will have to call more than 2 theatres?

d) Instead of each theatre deciding independently whether to show the film in 2D or 3D, suppose that the Seattle area gets exactly 6 copies of the film in 3D and 2 copies of the film in 2D. Which company gets which sort is decided at random. If I were to call only the four theatres closest to me, what is the probability that at least one of them would have received a 2D copy of the film?

Explanation / Answer

Answer to part a)

n = 8

p = 25

Mean = n*p = 8*0.25 = 2

Standard deviation = sqrt(n*p*q) = sqrt(8*0.25*0.75) = 1.2247

.

Answer to part b)

P(atleast 1) = 1 - P(none)

We use formula of binomial probability to find Porbability(none)

P(none) = 8C0 * (0.25)^0 * (0.75)^8 = 0.1001

P(atleast 1) = 1 - 0.1001 = 0.8999

.

Answer to part c)

P(1 theatre) = 0.25

P(2 theatres) = 0.75*0.25 = 0.1875

Thus P(more than 2 theatres) = 1 - P(1 theatre) - P(2 theatres)

P(more than 2 theatres) = 1 - 0.25-0.1875 = 0.5625

.

Answer to part d)

p = 2/8 = 0.25

n = 4

P(at least 1) = 1 - P(none)

Again, we make use of binomial probability formula to find the probability(none)

P(none) = 4C0 * (0.25)^0 * (0.75)^4 = 0.3164

Thus P(atleast 1) = 1 - 0.3164 = 0.6836

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